Preliminaries Main Theorems Constructions Hyperbolic geometry
Cusps of Hyperbolic Knots
Shruthi Sridhar
Cornell University [email protected]
Collaborators: Michael Moore, Rose Kaplan-Kelly, Brandon Shapiro, Joshua Wakefield
Research from SMALL 2016 Preliminaries Main Theorems Constructions Hyperbolic geometry
1 Preliminaries
2 Main Theorems
3 Constructions
4 Hyperbolic geometry Preliminaries Main Theorems Constructions Hyperbolic geometry
Hyperbolic Knots
Definition A hyperbolic knot or link is a knot or link whose complement in S3 is a 3-manifold that admits a complete hyperbolic structure.
This gives us a very useful invariant for hyperbolic knots: Volume (V) of the hyperbolic knot complement.
Figure 8 Knot 5 Chain Volume=2.0298... Volume=10.149..... Preliminaries Main Theorems Constructions Hyperbolic geometry
Cusp
Definition A Cusp of a knot K in S3 is defined as an open neighborhood of the knot intersected with the knot complement: S3 \ K
Thus, a cusp is a subset of the knot complement. Definition The Cusp Volume of a hyperbolic knot is the volume of the maximal cusp in the hyperbolic knot complement.
Since the cusp is a subset of the knot complement, the cusp volume of a knot is always less than or equal the volume of the knot complement. Fact √ The cusp volume of the figure-8 knot is 3
Preliminaries Main Theorems Constructions Hyperbolic geometry
Cusp Volume
The maximal cusp is the cusp of a knot expanded as much as possible until the cusp intersects itself. Preliminaries Main Theorems Constructions Hyperbolic geometry
Cusp Volume
The maximal cusp is the cusp of a knot expanded as much as possible until the cusp intersects itself. Definition The Cusp Volume of a hyperbolic knot is the volume of the maximal cusp in the hyperbolic knot complement.
Since the cusp is a subset of the knot complement, the cusp volume of a knot is always less than or equal the volume of the knot complement. Fact √ The cusp volume of the figure-8 knot is 3 Fact The optimal cusp volume of links will be achieved by first, maximizing a particular cusp till it touches itself, then a particular second cusp till it touches itself or the first cusp and so on...
Fact The optimal cusp volume√ of the minimally twisted 5-chain is 5 3√where√ the√ individual√ √ 3 3 3 3 cusps have volumes: 4 3, 4 , 4 , 4 , 4 respectively.
Preliminaries Main Theorems Constructions Hyperbolic geometry
Cusp Volume of Links
How should we expand cusps in a link to achieve maximal volume? Preliminaries Main Theorems Constructions Hyperbolic geometry
Cusp Volume of Links
How should we expand cusps in a link to achieve maximal volume? Fact The optimal cusp volume of links will be achieved by first, maximizing a particular cusp till it touches itself, then a particular second cusp till it touches itself or the first cusp and so on...
Fact The optimal cusp volume√ of the minimally twisted 5-chain is 5 3√where√ the√ individual√ √ 3 3 3 3 cusps have volumes: 4 3, 4 , 4 , 4 , 4 respectively. Preliminaries Main Theorems Constructions Hyperbolic geometry
Cusp Density
Definition CV Cusp Density (Dc ) of a knot or link is the ratio: V where CV is the total cusp volume and V is the hyperbolic volume of the complement.
Volume=2.0298...√ Volume=10.149...√ Cusp Volume= 3 Cusp Volume = 5 3 Cusp Density=0.853 Cusp Density=0.853 Theorem (SMALL 2016) For any x ∈ [0, 0.853...], there exist hyperbolic link complements with cusp density arbitrarily close to x.
Theorem (SMALL 2016) For any x ∈ [0, 0.6826...], there exist hyperbolic knot complements with cusp density arbitrarily close to x.
Preliminaries Main Theorems Constructions Hyperbolic geometry
Main Theorems
Fact (B¨or¨oczky 1978) The highest cusp density a hyperbolic manifold can have is 0.853..., the cusp density of the figure-8 knot and the minimally twisted 5-chain. Preliminaries Main Theorems Constructions Hyperbolic geometry
Main Theorems
Fact (B¨or¨oczky 1978) The highest cusp density a hyperbolic manifold can have is 0.853..., the cusp density of the figure-8 knot and the minimally twisted 5-chain.
Theorem (SMALL 2016) For any x ∈ [0, 0.853...], there exist hyperbolic link complements with cusp density arbitrarily close to x.
Theorem (SMALL 2016) For any x ∈ [0, 0.6826...], there exist hyperbolic knot complements with cusp density arbitrarily close to x. Preliminaries Main Theorems Constructions Hyperbolic geometry
Constructions
Three constructions provide essential elements to the proofs: Dehn Filling Cyclic covers Belted sums Cq C Fact
As q → ∞, the Volume and Cusp T T volume approaches that of the original link before Dehn filling Vq V
Preliminaries Main Theorems Constructions Hyperbolic geometry
Dehn Filling
Definition (1, q) Dehn filling on an unknotted component of a link complement gives a link complement of the original link, without the trivial component, and the strands through it twisted q times. Preliminaries Main Theorems Constructions Hyperbolic geometry
Dehn Filling
Definition (1, q) Dehn filling on an unknotted component of a link complement gives a link complement of the original link, without the trivial component, and the strands through it twisted q times.
Cq C Fact
As q → ∞, the Volume and Cusp T T volume approaches that of the original link before Dehn filling Vq V We know that volume of an n−fold cover Ln is n times the volume of the original link L.
V (Ln) = nV (L) In fact, the same holds for cusp volumes and we can see this as a multiplication of the fundamental domain.
CV (Ln) = nCV (L)
Preliminaries Main Theorems Constructions Hyperbolic geometry n-fold Covers
The following construction is taking ”n-fold cyclic covers” of a tangle across a twice punctured disc.
Twisted Borromean Rings 3-fold cover Preliminaries Main Theorems Constructions Hyperbolic geometry n-fold Covers
The following construction is taking ”n-fold cyclic covers” of a tangle across a twice punctured disc.
Twisted Borromean Rings 3-fold cover
We know that volume of an n−fold cover Ln is n times the volume of the original link L.
V (Ln) = nV (L) In fact, the same holds for cusp volumes and we can see this as a multiplication of the fundamental domain.
CV (Ln) = nCV (L) V (L1) + V (L2) = V (L1+2)
We know that volume of a belted sum is the sum of the volumes of the original 2 links. What happens to cusp volumes?
Preliminaries Main Theorems Constructions Hyperbolic geometry
Belted Sums
The following construction is taking a belted sum of links cross a twice punctured disc
T2 T1 T2
T1
L1 L2 Belted Sum: L1+2 Preliminaries Main Theorems Constructions Hyperbolic geometry
Belted Sums
The following construction is taking a belted sum of links cross a twice punctured disc
T2 T1 T2
T1
L1 L2 Belted Sum: L1+2
V (L1) + V (L2) = V (L1+2)
We know that volume of a belted sum is the sum of the volumes of the original 2 links. What happens to cusp volumes? The boundary of a cusp is a torus. The meridian is marked in yellow and the longitude in red.
Preliminaries Main Theorems Constructions Hyperbolic geometry
Belted Sums
T2
Cusp volume doesn’t add T1 T2 T so nicely. 1
L1 L2 L1+2 ∗ Theorem (SMALL 16) ∗Under the absence of poking 2 m1 RCV (L1+2) = RCV (L1) + RCV (L2) m2
m1 and m2 are meridian lengths of the maximal cusps of L1 and L2 Preliminaries Main Theorems Constructions Hyperbolic geometry
Belted Sums
T2
Cusp volume doesn’t add T1 T2 T so nicely. 1
L1 L2 L1+2 ∗ Theorem (SMALL 16) ∗Under the absence of poking 2 m1 RCV (L1+2) = RCV (L1) + RCV (L2) m2
m1 and m2 are meridian lengths of the maximal cusps of L1 and L2
The boundary of a cusp is a torus. The meridian is marked in yellow and the longitude in red. Preliminaries Main Theorems Constructions Hyperbolic geometry
Belted Sums
Heuristic Proof: Preliminaries Main Theorems Constructions Hyperbolic geometry
Belted Sums
Heuristic Proof: Preliminaries Main Theorems Constructions Hyperbolic geometry
Belted Sums
Heuristic Proof: Preliminaries Main Theorems Constructions Hyperbolic geometry
Belted Sums
Heuristic Proof:
Volume of Cusp ∝ meridian2
2 new m1 RCV (L2 ) = RCV (L2) m2 2 m1 RCV (L1+2) = RCV (L1) + RCV (L2) m2 2 m1 RCV = aRCV1 + b RCV2 and V = aV1 + bV2 m2 Finally, perform high Dehn filling to the unknotted component to 2 a m1 ( )RCV1+ RCV2 b m2 get a knot with cusp density close to a . ( b )V1+V2 Choosing the right links gives us cusp density dense in [0, 0.6826..]
Preliminaries Main Theorems Constructions Hyperbolic geometry
Finishing the proof
We find two tangles augmented by a link with volumes V1 and V2, restricted cusp volume RCV1 and RCV2 and meridians m1, m2. We take 0a0 fold covers of one and belt sum with a 0b0 fold cover of the other to get a link. Preliminaries Main Theorems Constructions Hyperbolic geometry
Finishing the proof
We find two tangles augmented by a link with volumes V1 and V2, restricted cusp volume RCV1 and RCV2 and meridians m1, m2. We take 0a0 fold covers of one and belt sum with a 0b0 fold cover of the other to get a link.
2 m1 RCV = aRCV1 + b RCV2 and V = aV1 + bV2 m2 Finally, perform high Dehn filling to the unknotted component to 2 a m1 ( )RCV1+ RCV2 b m2 get a knot with cusp density close to a . ( b )V1+V2 Choosing the right links gives us cusp density dense in [0, 0.6826..] Fact Cusps of hyperbolic knots lift to disjoint unions of horoballs in the upper half model of hyperbolic space
Preliminaries Main Theorems Constructions Hyperbolic geometry
Covering space H3
Fact Complements of hyperbolic knots lift to several copies of 3 polyhedra in H Preliminaries Main Theorems Constructions Hyperbolic geometry
Covering space H3
Fact Complements of hyperbolic knots lift to several copies of 3 polyhedra in H
Fact Cusps of hyperbolic knots lift to disjoint unions of horoballs in the upper half model of hyperbolic space We first look at the boundary of the cusp. When we cut along the meridian and along the longitude, we get a rectangle with the identifications as shown. This corresponds to a fundamental domain on the boundary of the horoball centered at ∞.
Preliminaries Main Theorems Constructions Hyperbolic geometry
Cusp Diagrams
Where do different parts of the cusp go in this horoball diagram? Preliminaries Main Theorems Constructions Hyperbolic geometry
Cusp Diagrams
Where do different parts of the cusp go in this horoball diagram?
We first look at the boundary of the cusp. When we cut along the meridian and along the longitude, we get a rectangle with the identifications as shown. This corresponds to a fundamental domain on the boundary of the horoball centered at ∞. When the cusp is maximized
Preliminaries Main Theorems Constructions Hyperbolic geometry
Cusp Diagrams-Example of figure 8
View from ball at infinity Preliminaries Main Theorems Constructions Hyperbolic geometry
Cusp Diagrams-Example of figure 8
View from ball at infinity
When the cusp is maximized Preliminaries Main Theorems Constructions Hyperbolic geometry
Horoball Diagrams of Links
3 Consider the Borromean Rings. The 3 cusps lift to horoballs in H . Each horoball (tangent to z = 0 or centered at ∞) is actually centered at the same point at infinity. This gives us a choice to view the horoball diagram from a particular horoball centered at infinity in the diagram (the one that looks like space enclosed above a plane).
Horoball Diagram The Borromean Cusp Diagram looking Rings from the blue cusp at infinity A theorem by Adams states that a twice punctured disc in a hyperbolic 3-manifold always lifts to a totally geodesic surface in the upper half space, as shown.
P3
1 longitude 1 1 1
2 2 2 2
P1 P2 Twice Punctured Disc Horoball Diagram
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Links and Twice Punctured Discs
Consider a trivial component around 2 strands. This bounds a twice A 1 2 punctured disc in the complement as P Q shown. 1 longitude 1
2 2
Horoball Diagram
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Links and Twice Punctured Discs
Consider a trivial component around 2 strands. This bounds a twice A 1 2 punctured disc in the complement as P Q shown.
A theorem by Adams states that a twice punctured disc in a hyperbolic 3-manifold always lifts to a totally geodesic surface in the upper half space, as shown.
P3
1 1
2 2
P1 P2 Twice Punctured Disc Preliminaries Main Theorems Constructions Hyperbolic geometry
Links and Twice Punctured Discs
Consider a trivial component around 2 strands. This bounds a twice A 1 2 punctured disc in the complement as P Q shown.
A theorem by Adams states that a twice punctured disc in a hyperbolic 3-manifold always lifts to a totally geodesic surface in the upper half space, as shown.
P3
1 longitude 1 1 1
2 2 2 2
P1 P2 Twice Punctured Disc Horoball Diagram Preliminaries Main Theorems Constructions Hyperbolic geometry
Twice Punctured Discs in horoball Diagrams
Twisted Borromean Cusp Diagram of the twice Rings punctured looking from the blue cusp
The twice punctured disc, when viewed from the blue cusp at infinity surfaces can be seen as passing through the longitude of the blue cusp in the horoball diagram. Preliminaries Main Theorems Constructions Hyperbolic geometry
Horoball diagram of Belted Sums Preliminaries Main Theorems Constructions Hyperbolic geometry
Acknowledgements
I would like to Professor Colin Adams for his mentorship, advice and motivation, the Williams College Science Center and NSF REU grant DMS-1347804 for funding, and my fellow students in the Knot Group: Josh, Rosie, Michael and Brandon. Preliminaries Main Theorems Constructions Hyperbolic geometry
References
Colin Adams (1985) Thrice-punctured spheres in hyperbolic 3-manifolds Trans. AMS, 287 (2), pp. 645-656 Colin Adams, Aaron Calderon, Xinyi Jiang, Alexander Kastner, Gregory Kehne, Nathaniel Mayer, Mia Smith. (2015) Volume and determinant densities of Hyperbolic Links http://arxiv.org/pdf/1510.06050.pdf B¨or¨oczky, K (1978). Packing of spheres in spaces of constant curvature Acta Mathematica Academiae Scientiarum Hungaricae 32: 243-261. Marc Culler, Nathan M Dunfield, Matthias Goerner, and Jeffrey R Weeks. SnapPy a computer program for studying the geometry and topology of 3-manifolds. http://snappy.computop.org